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Theorem prfcl 17197
Description: The pairing of functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐷 is a functor 𝐹, 𝐺⟩:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfcl.p 𝑃 = (𝐹 ⟨,⟩F 𝐺)
prfcl.t 𝑇 = (𝐷 ×c 𝐸)
prfcl.c (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prfcl.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
Assertion
Ref Expression
prfcl (𝜑𝑃 ∈ (𝐶 Func 𝑇))

Proof of Theorem prfcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfcl.p . . . 4 𝑃 = (𝐹 ⟨,⟩F 𝐺)
2 eqid 2826 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2826 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
4 prfcl.c . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 prfcl.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
61, 2, 3, 4, 5prfval 17193 . . 3 (𝜑𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
7 fvex 6447 . . . . . . 7 (Base‘𝐶) ∈ V
87mptex 6743 . . . . . 6 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
97, 7mpt2ex 7511 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
108, 9op1std 7439 . . . . 5 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
116, 10syl 17 . . . 4 (𝜑 → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
128, 9op2ndd 7440 . . . . 5 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (2nd𝑃) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
136, 12syl 17 . . . 4 (𝜑 → (2nd𝑃) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
1411, 13opeq12d 4632 . . 3 (𝜑 → ⟨(1st𝑃), (2nd𝑃)⟩ = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
156, 14eqtr4d 2865 . 2 (𝜑𝑃 = ⟨(1st𝑃), (2nd𝑃)⟩)
16 prfcl.t . . . . 5 𝑇 = (𝐷 ×c 𝐸)
17 eqid 2826 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
18 eqid 2826 . . . . 5 (Base‘𝐸) = (Base‘𝐸)
1916, 17, 18xpcbas 17172 . . . 4 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
20 eqid 2826 . . . 4 (Hom ‘𝑇) = (Hom ‘𝑇)
21 eqid 2826 . . . 4 (Id‘𝐶) = (Id‘𝐶)
22 eqid 2826 . . . 4 (Id‘𝑇) = (Id‘𝑇)
23 eqid 2826 . . . 4 (comp‘𝐶) = (comp‘𝐶)
24 eqid 2826 . . . 4 (comp‘𝑇) = (comp‘𝑇)
25 funcrcl 16876 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
264, 25syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2726simpld 490 . . . 4 (𝜑𝐶 ∈ Cat)
2826simprd 491 . . . . 5 (𝜑𝐷 ∈ Cat)
29 funcrcl 16876 . . . . . . 7 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
305, 29syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
3130simprd 491 . . . . 5 (𝜑𝐸 ∈ Cat)
3216, 28, 31xpccat 17184 . . . 4 (𝜑𝑇 ∈ Cat)
33 relfunc 16875 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
34 1st2ndbr 7480 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3533, 4, 34sylancr 583 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
362, 17, 35funcf1 16879 . . . . . . . 8 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
3736ffvelrnda 6609 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
38 relfunc 16875 . . . . . . . . . 10 Rel (𝐶 Func 𝐸)
39 1st2ndbr 7480 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
4038, 5, 39sylancr 583 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
412, 18, 40funcf1 16879 . . . . . . . 8 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
4241ffvelrnda 6609 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
43 opelxpi 5380 . . . . . . 7 ((((1st𝐹)‘𝑥) ∈ (Base‘𝐷) ∧ ((1st𝐺)‘𝑥) ∈ (Base‘𝐸)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
4437, 42, 43syl2anc 581 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
4544fmpttd 6635 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
4611feq1d 6264 . . . . 5 (𝜑 → ((1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)) ↔ (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸))))
4745, 46mpbird 249 . . . 4 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
48 eqid 2826 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
49 ovex 6938 . . . . . . 7 (𝑥(Hom ‘𝐶)𝑦) ∈ V
5049mptex 6743 . . . . . 6 ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) ∈ V
5148, 50fnmpt2i 7503 . . . . 5 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) Fn ((Base‘𝐶) × (Base‘𝐶))
5213fneq1d 6215 . . . . 5 (𝜑 → ((2nd𝑃) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) Fn ((Base‘𝐶) × (Base‘𝐶))))
5351, 52mpbiri 250 . . . 4 (𝜑 → (2nd𝑃) Fn ((Base‘𝐶) × (Base‘𝐶)))
54 eqid 2826 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
5535adantr 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
56 simprl 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
57 simprr 791 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
582, 3, 54, 55, 56, 57funcf2 16881 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
5958ffvelrnda 6609 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐹)𝑦)‘) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
60 eqid 2826 . . . . . . . . . 10 (Hom ‘𝐸) = (Hom ‘𝐸)
6140adantr 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
622, 3, 60, 61, 56, 57funcf2 16881 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
6362ffvelrnda 6609 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
64 opelxpi 5380 . . . . . . . 8 ((((𝑥(2nd𝐹)𝑦)‘) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ ((𝑥(2nd𝐺)𝑦)‘) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
6559, 63, 64syl2anc 581 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
664adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
675adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐶 Func 𝐸))
681, 2, 3, 66, 67, 56prf1 17194 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
691, 2, 3, 66, 67, 57prf1 17194 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) = ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩)
7068, 69oveq12d 6924 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(Hom ‘𝑇)⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩))
7137adantrr 710 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
7242adantrr 710 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
7336ffvelrnda 6609 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
7473adantrl 709 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
7541ffvelrnda 6609 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
7675adantrl 709 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
7716, 17, 18, 54, 60, 71, 72, 74, 76, 20xpchom2 17180 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(Hom ‘𝑇)⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
7870, 77eqtrd 2862 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
7978adantr 474 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
8065, 79eleqtrrd 2910 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
8180fmpttd 6635 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
8213oveqd 6923 . . . . . . 7 (𝜑 → (𝑥(2nd𝑃)𝑦) = (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦))
8348ovmpt4g 7044 . . . . . . . 8 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) ∈ V) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
8450, 83mp3an3 1580 . . . . . . 7 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
8582, 84sylan9eq 2882 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
8685feq1d 6264 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) ↔ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦))))
8781, 86mpbird 249 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
88 eqid 2826 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
8935adantr 474 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
90 simpr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
912, 21, 88, 89, 90funcid 16883 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
92 eqid 2826 . . . . . . 7 (Id‘𝐸) = (Id‘𝐸)
9340adantr 474 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
942, 21, 92, 93, 90funcid 16883 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st𝐺)‘𝑥)))
9591, 94opeq12d 4632 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)), ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))⟩ = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
964adantr 474 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝐷))
975adantr 474 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐸))
9827adantr 474 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
992, 3, 21, 98, 90catidcl 16696 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
1001, 2, 3, 96, 97, 90, 90, 99prf2 17196 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝐶)‘𝑥)) = ⟨((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)), ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))⟩)
1011, 2, 3, 96, 97, 90prf1 17194 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
102101fveq2d 6438 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘((1st𝑃)‘𝑥)) = ((Id‘𝑇)‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
10328adantr 474 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
10431adantr 474 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
10516, 103, 104, 17, 18, 88, 92, 22, 37, 42xpcid 17183 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
106102, 105eqtrd 2862 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘((1st𝑃)‘𝑥)) = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
10795, 100, 1063eqtr4d 2872 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝑇)‘((1st𝑃)‘𝑥)))
108 eqid 2826 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
109353ad2ant1 1169 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
110 simp21 1269 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
111 simp22 1270 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
112 simp23 1271 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
113 simp3l 1264 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
114 simp3r 1265 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
1152, 3, 23, 108, 109, 110, 111, 112, 113, 114funcco 16884 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)))
116 eqid 2826 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
11753ad2ant1 1169 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐺 ∈ (𝐶 Func 𝐸))
11838, 117, 39sylancr 583 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
1192, 3, 23, 116, 118, 110, 111, 112, 113, 114funcco 16884 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓)))
120115, 119opeq12d 4632 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)), ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))⟩ = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
12143ad2ant1 1169 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐹 ∈ (𝐶 Func 𝐷))
122273ad2ant1 1169 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
1232, 3, 23, 122, 110, 111, 112, 113, 114catcocl 16699 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
1241, 2, 3, 121, 117, 110, 112, 123prf2 17196 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ⟨((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)), ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))⟩)
1251, 2, 3, 121, 117, 110prf1 17194 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
1261, 2, 3, 121, 117, 111prf1 17194 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑦) = ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩)
127125, 126opeq12d 4632 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩ = ⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩)
1281, 2, 3, 121, 117, 112prf1 17194 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑧) = ⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)
129127, 128oveq12d 6924 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧)) = (⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩))
1301, 2, 3, 121, 117, 111, 112, 114prf2 17196 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝑃)𝑧)‘𝑔) = ⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩)
1311, 2, 3, 121, 117, 110, 111, 113prf2 17196 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑦)‘𝑓) = ⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩)
132129, 130, 131oveq123d 6927 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = (⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩(⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩))
133363ad2ant1 1169 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
134133, 110ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
135413ad2ant1 1169 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
136135, 110ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
137133, 111ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
138135, 111ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
139133, 112ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
140135, 112ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐸))
1412, 3, 54, 109, 110, 111funcf2 16881 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
142141, 113ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
1432, 3, 60, 118, 110, 111funcf2 16881 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
144143, 113ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
1452, 3, 54, 109, 111, 112funcf2 16881 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
146145, 114ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
1472, 3, 60, 118, 111, 112funcf2 16881 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐸)((1st𝐺)‘𝑧)))
148147, 114ffvelrnd 6610 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑔) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐸)((1st𝐺)‘𝑧)))
14916, 17, 18, 54, 60, 134, 136, 137, 138, 108, 116, 24, 139, 140, 142, 144, 146, 148xpcco2 17181 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩(⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
150132, 149eqtrd 2862 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
151120, 124, 1503eqtr4d 2872 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)))
1522, 19, 3, 20, 21, 22, 23, 24, 27, 32, 47, 53, 87, 107, 151isfuncd 16878 . . 3 (𝜑 → (1st𝑃)(𝐶 Func 𝑇)(2nd𝑃))
153 df-br 4875 . . 3 ((1st𝑃)(𝐶 Func 𝑇)(2nd𝑃) ↔ ⟨(1st𝑃), (2nd𝑃)⟩ ∈ (𝐶 Func 𝑇))
154152, 153sylib 210 . 2 (𝜑 → ⟨(1st𝑃), (2nd𝑃)⟩ ∈ (𝐶 Func 𝑇))
15515, 154eqeltrd 2907 1 (𝜑𝑃 ∈ (𝐶 Func 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  Vcvv 3415  cop 4404   class class class wbr 4874  cmpt 4953   × cxp 5341  Rel wrel 5348   Fn wfn 6119  wf 6120  cfv 6124  (class class class)co 6906  cmpt2 6908  1st c1st 7427  2nd c2nd 7428  Basecbs 16223  Hom chom 16317  compcco 16318  Catccat 16678  Idccid 16679   Func cfunc 16867   ×c cxpc 17162   ⟨,⟩F cprf 17165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-ixp 8177  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-3 11416  df-4 11417  df-5 11418  df-6 11419  df-7 11420  df-8 11421  df-9 11422  df-n0 11620  df-z 11706  df-dec 11823  df-uz 11970  df-fz 12621  df-struct 16225  df-ndx 16226  df-slot 16227  df-base 16229  df-hom 16330  df-cco 16331  df-cat 16682  df-cid 16683  df-func 16871  df-xpc 17166  df-prf 17169
This theorem is referenced by:  prf1st  17198  prf2nd  17199  uncfcl  17229  uncf1  17230  uncf2  17231  yonedalem1  17266  yonedalem21  17267  yonedalem22  17272
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